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Evaporative Cooling of 87Rb with Microwave Radiation
Christian Prosko
Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1
(Dated December 3, 2015)
An evaporative cooling system is employed via microwave radiation, incident on 87Rb contained in a
magnetic quadrupole trap, to drive transitions in the atomic vapor between trapped and untrapped states.
In addition, the degree of sample losses due to these transitions is studied with consideration given to
the amplitude and frequency of incident electromagnetic fields. To understand the underlying physics of
evaporative cooling, a brief introduction to modern cooling techniques and theory is given.
I. INTRODUCTION
Ultracold atoms posess a plethora of interesting char-
acteristics, warranting the torrent of recent investigation
into the subject. For example: superfluids exhibit zero
viscosity, while superconducters have no electrical resis-
tance and expel magnetic field lines. Another instance
of this is Bose-Einstein condensation (BEC), existing at
low enough temperatures such that every particle within
a boson gas ‘condenses’ into the quantum ground state.
Such an exotic state of matter is characterized by a
minimal deviation of the sample’s velocity distribution,
so that the Heisenberg Uncertainty Principle necessitates
a large spread in the wave-function of every boson. This
constitutes an intrinsically quantum mechanical state,
that is, a gas of entangled bosons describable only by
a single multi-particle wave-function.
The challanges inherent in reaching the low-
temperature limit of quantum degeneracy, wherein
a sample’s DeBroglie wavelength is on the order of the
inter-particle spacing, is illustrated in the gap between
the discovery of superconductivity in 1911 at 4 Kelvin
[1] and the first Bose-Einstein condensate observed in a
dilute gas of 87Rb at 170nK, eighty-four years later [2].
Doppler cooling proved an effective method for bringing
the temperature of atomic vapors down to microkelvin
temperatures, but further cooling was impossible without
another method.
Thusly, we design and implement an apparatus for
the purpose of cooling a gas of 87Rb from this limit
down to the nanokelvin range. One of the most popular
methods of doing so, and the one utilized here, is that of
evaporative cooling using microwave radiation. Just as
blowing on a cup of coffee tends to cool it by allowing the
hottest water molecules to evaporate, this method hinges
on providing a sort of ‘exit route’ for the hottest atoms
in a trap, while holding on to the colder atoms.
II. THEORY & METHODOLOGY
A. Doppler Cooling
Consider monochromatic laser light of frequency ω,
incident on an effectively non-interacting atomic gas,
carrying momentum h~k. If one of these atoms has velocity
~v, then in said particle’s rest frame, the light is Doppler-
shifted to frequency ωD = −~k·~v. Due to the discretization
of electronic energy levels, a photon may transfer its
momentum to the atom only if its energy Eph = hω so
that its frequency is in resonance with one such level.
The Doppler shifting described above, however, makes
this resonance dependent on the atomic velocity.
It is thus clear how lasers may be applied to reduce gas
temperature. If a laser is tuned to just below a transition
energy, it will only transfer its momentum to atoms
travelling in the opposite direction of light propagation.
Multiple pairs of opposing lasers directed radially inward
toward a sample could then be made to reduce the average
kinetic energy of a sample [3, p. 73-122].
Unfortunately, Doppler cooling has a limit. The ef-
fect of spontaneous emission of photons is on average
isotropic, so that no net momentum is imparted to the gas
through this process. On the other hand, it increases the
mean square velocity of constituents, and thus imparts
heat to the sample. At the Doppler temperature TD,
given by [3, p. 57]:
TD =hγ
2kB(1)
the cooling effect is overwhelmed by spontaneous emis-
sion1. Here γ is the natural linewidth of the excited
1Some more complicated laser cooling methods such as Sisyphus
cooling [4] use an optical polarization gradient to cool below this
limit, but they still have limits above the temperature range
for quantum degeneracy around the recoil temperature: Tr =
(hk)2/(2kBM).
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Christian Prosko (1353554) PHYS 499 - Final Report
state, i.e. the reciprocal of the excited state’s typical
lifetime. Generally, this limit is on the order of 100µK;
this is several orders of magnitude above the temperature
required for BEC. Hence, we are motivated to study
a complementary method of cooling utilizing microwave
radiation.
B. Magnetic Trapping of Neutral Atoms
As a result of the Zeeman effect, a magnetic field ~B
introduces a term Hm = −µ · ~B (µ being the magnetic
dipole operator) to the single-atom Hamiltonian, result-
ing in a splitting of hyperfine energy levels. For alkali
atoms which have a single valence electron in an s-orbital,
hydrogenic quantum states are effective in describing
their properties, since they may be approximated as
a heavy nucleus of +e charge surrounded by a single
electron. In a state with zero orbital electron momentum
these splittings ∆EF may be calculated from the Breit-
Rabi formula, as derived in Appendix A:
∆E|F=I±1/2,mF 〉 = − ∆Ehf4(I + 1/2)
+ µBgImFB
± ∆Ehf2
√1 +
2mFx
I + 1/2+ x2 (2)
In which x ≡ gJµBB∆Ehf
, ∆Ehf denotes the hyperfine energy
gaps in the absence of an external magnetic field, gI , gJ ,
and gF are Lande g-factors2, and mF ∈ −F,−F +
1, ..., F − 1, F is the magnetic quantum number. En-
ergy splittings calculated for 87Rb from this formula are
plotted in Figure 1.
The most important consequence of eq. (2) is that
certain states have a lower energy in higher magnetic
fields, whereas other |F,mF 〉 states seek lower magnetic
field strength to minimize their energy or are unaffected
by this added potential. Thusly, a magnetic field with a
region of minimum strength may trap ‘high-field seeking’
states, while allowing others to escape (see Figure 1).
This is more readily apparent in the weak-field limit of
Zeeman splittings, where the static field may be treated
as a small perturbation of the fine structure Hamiltonian
for which |F,mF 〉 are the eigenstates. The first order
energy correction is then:
2These are constants for a given isotope corresponding to nuclear
spin I, total electron angular momentum J = L+ S (where L and
S are orbital and spin angular momentum for the valence electron),
and total atomic angular momentum F = I + J .
FIG. 1. Ground-state energy splitting of hyperfine levels in87Rb in the presence of a uniform magnetic field, calculated
from the Breit-Rabi formula.
∆E|F,mF 〉 = 〈F,mF | µ · ~B |F,mF 〉
= 〈F,mF |(gFµB
h
)F · ~B |F,mF 〉
=gFµBB
h〈F,mF | Fz |F,mF 〉
= gFµBmFB (3)
noting that gF = −1/2,1/2 for F = 1, 2 respectively, and
that z is the axis of quantization i.e. the direction of ~B.
This is an application of the theorem which states that if
degenerate eigenfunctions of an unperturbed Hamiltonian
H0 are also eigenfunctions with distinct eigenvalues of a
Hermitian operator A which commutes with both H0 and
the perturbation H ′, then the first order energy correction
is the same as that from non-degenerate perturbation
theory. In this case, A = Fz, which commutes with the
fine structure Hamiltonian and (on time average) with
Hm [5, p. 277].
To achieve the sort of trapping field described previ-
ously, a magnetic quadrupole trap is used. By assembling
three Helmholtz pairs along perpendicular axes, a mag-
netic field is established with a region in the center of the
form3:
3The factor of 2 in the z-component is necessitated by Maxwell’s
equation: ~∇ · ~B = 0. We use this to our advantage, pointing the
z-axis vertically to reduce any gravitational effects on our sample.
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Christian Prosko (1353554) PHYS 499 - Final Report
FIG. 2. Hyperfine energy levels in a linear magnetic field:~B = (0.5Tm-1)(xx+ yy − 2zz) at constant x = 5cm.
~B = B0(xx+ yy − 2zz) (4)
Substituting this field into eq. (2), we find that ~r = 0
is either a minimum or maximum in potential energy
depending on the magnetic quantum number (see fig. 2).
This sort of magnetic trapping scheme has a significant
downside, however. At the point where magnetic field
strength vanishes, the Zeeman energies beome degener-
ate, making the spin (i.e. mF value) ambiguous. Once
the atom reemerges into a region of non-zero magnetic
field, its spin may have flipped in what is called a ‘Ma-
jorana transition’ [6]. These transitions occur frequently
enough that resulting particle losses inhibit trap lifetime
to a detrimental degree. In addition, the experiments
possible in a magnetic trap are limited outside of Bose-
Einstein Condensation, since whether an atom is held
in the trap or not is fundamentally dependent on the
internal (hyperfine) atomic state.
For these reasons, magnetic quadrupole traps are gen-
erally supplemented with a perturbative light source to
create a bias in the magnetic field zero. A common
example of this is a time-oscillating magnetic field de-
signed to translate the zero-point around in a circle.
For frequencies greater than the atomic orbital frequency
around the field zero, low-field seeking atoms ‘follow’ the
zero-point without reaching it, effectively eliminating the
possibility of spin-flip transitions [7].
Another such preventative technique – the one used
by this group – is an optical dipole trap directed about
the magnetic trap minimum. These traps have been
well established and described (for example, see: [3, 8]),
and operate by using lasers to simultaneously induce an
oscillating electric dipole moment in atoms, whilst using
the resulting dipole force to pull the atoms to the region
where laser light is most focused (this is called the beam
waist). This force results from a gradiant in the Stark
shift4 due to spatial variation in the intensity of focused
light. As a result, atoms are trapped in the vicinity
of the beam waist, where the electric field gradient is
strongest. In conjunction with a magnetic quadrupole
trap, optical dipole traps can produce shallow minima in
potential energy surrounding the point of zero magnetic
field, encouraging trapped atoms to avoid the point of
vanishing field strength.
C. Rabi Flopping
Evaporative cooling relies on electromagnetic radiation
to instigate transitions between magnetic sublevels in
atoms, so it is worth discussing the manner in which
a monochromatic light source accomplishes such a task.
In general, Rabi Flopping describes a system’s evolution
between two otherwise stationary states as a result of a
perturbative time-oscillating potential5. In our case, we
approximate the unperturbed system as being a rubidium
atom confined to an approximately uniform region of the
magnetic trapping field, driven to oscillate between F = 1
and F = 2 states by a linearly polarized magnetic wave~Bmw of microwave frequency near the ω0 ≡ 6.834GHz
transition [11]. The trapping field defines a quantization
axis (say, along the z axis) and thus creates a magnetic
dipole moment µ = µI+µS+µL, so that the perturbative
potential is:
V = ~µ · ~Bmw = (µI + µS) · ~Bmw (5)
in the ground state of rubidium (L = 0). Before con-
tinuing, it is worth noting that since the trapping field’s
directionality varies with position, microwave radiation
will have a distinctly varying effect at certain regions in
the trap. In particular, when z happens to be exactly
parallel or perpendicular to ~Bmw. This implies that
4The energy shifting effect of electrons being pulled in the opposite
direction as the positive nuclei in an electric field.5Most graduate quantum physics texts cover this topic; see for
example: [3, 9, 10].
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Christian Prosko (1353554) PHYS 499 - Final Report
any usage of microwaves to manipulate energy states of
trapped atoms requires a consideration of the system’s
geometry. Since evaporative cooling relies on transitions
from trapped to untrapped states, only the |2, 2〉 ↔|1, 1〉 and |2, 1〉 ↔ |1, 1〉 transitions are relevant to our
discussion6. Thusly, we suppose ~Bmw = Bmw cos (ωt)x is
perpendicular to the trapping field in some region, so:
V =µBB
mw
h(gI Ix + gSSx) cosωt (6)
Obviously, this is not a two-state system in the |F,mF 〉basis, but considering the case of a |2, 2〉 → |1, 1〉transition, where the microwaves are turned on at t = 0
and remain on, we may say that the general quantum
state for t < 0 is |ψ〉 = c1 |1, 1〉+ c2eiω0t |2, 2〉 since both
states are eigenstates of the unperturbed Hamiltonian.
Here we have shifted our zero energy such that E1 = 0
and E2 = hω0. After V is introduced at t = 0, we assume
that our state is invariant except for a time dependence
granted to the previously constant coefficients c1 and c2.
The Schrodinger equation then yields:
(c′1(t)
c′2(t)eiω0t
)=eiωt + e−iωt
2i
(0 Ω
Ω 0
)(c1(t)
c2(t)eiω0t
)(7)
where Ω ≡ 〈2, 2| V |1, 1〉 /(h cos (ωt)) is a constant
called the Rabi Frequency. Note that one of the time
derivative terms cancels the unperturbed Hamiltonian
terms, and that the diagonal potential terms are 0 [12].
Clearly, for near-resonance ω ≈ ω0, some terms oscillate
at almost twice ω0, while others oscillate very slowly
(frequency ω0 − ω). We thus make the rotating wave
approximation: That these rapidly oscillating terms are
neligible [13]. Then:
ih
(c′1(t)
c′2(t)
)=
(0 1
2Ω12Ω 0
)(c1(t)
c2(t)
)(8)
Solving for one coefficient in either equation, differen-
tiating the other and substituting in the result, we have:
c′′1(t)− i(ω0 − ω)c′1(t) +1
4Ω2c1(t) = 0 (9a)
6Transitions between mF levels at constant F can also untrap
atoms, but the corresponding transition frequencies are far below
the microwave range. Radio-frequency evaporative cooling, for
example, uses such transitions.
c′′2(t) + i(ω0 − ω)c′2(t) +1
4Ω2c2(t) = 0 (9b)
with γ ≡ ω0 − ω being the frequency detuning. This
differential equation can be solved analytically, simply by
choosing integration factors such that c1(t) = b1(t)e12 iγt
and c2 = b2(t)e−12 iγt for some functions b1(t), b2(t). Our
differential equations then become:
b′′1(t)− 1
2iγb′1(t) +
1
4(γ2 + Ω2)b1(t) = 0 (10a)
b′′2(t) +1
2iγb′2(t) +
1
4(γ2 + Ω2)b2(t) = 0 (10b)
These equations are merely those for simple har-
monic oscillators, and have general solutions of the form
A sin Ω′t+B cos Ω′t, where one coefficient is constrained
by the first derivative term and Ω′ ≡ 12
√Ω2 + γ2. Sup-
posing our system is initially prepared in the |2, 2〉 state
such that c1(0) = 0 and c2(0) = 1, these constraints in
addition to the total probability condition |c1|2+|c2|2 = 1
provide the final solution:
c1(t) = −i Ω
Ω′sin (Ω′t)e−
12 iγt (11a)
c2(t) =
[cos (Ω′t)− i Ω
Ω′sin (Ω′t)
]e
12 iγt (11b)
Qualitatively, the most important consequences of this
result are that detuned light may still initiate transitions
between levels (albeit with lower probability), and the
probability of finding a particle in |2, 2〉 or |1, 1〉 oscillates
with frequency Ω′. The value of Ω has been calculated
in [12]7 and in Appendix B, so that the frequency of
oscillation between states is:
Ω′ =1
2
√(5.81945 · 1021Hz2/T2)Bmw2 + γ2 (12)
D. Evaporative Cooling
Finally, we are prepared to discuss evaporative cooling.
Supposing all or most of the atoms in a Rubidium-87
7This calculation involves expanding |F,mF 〉 states in terms of
|mI ,mJ 〉 states and calculating the expectation value using
Clebsch-Gordan coefficients. µI is neglected in [12], but this is
reasonable since gI ≈ −0.001 is several orders of magnitude smaller
than gS ≈ 2 [11].
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Christian Prosko (1353554) PHYS 499 - Final Report
sample have been prepared and magnetically trapped
in the |F,mF 〉 = |2, 2〉 state, (say, via optical pumping
[14]) and that a microwave source of tunable frequency
near the 6.834GHz transition from F = 2 → F = 1 is
incident upon the magnetically trapped atoms. Only
the |2, 2〉 → |1, 1〉 transition can untrap these atoms
whilst also lying in the microwave range of transition
frequencies. In fact, this is the only possible transition
from |2, 2〉 that could be driven by microwave frequencies,
since transitions involving ∆mF > 1 are impossible
regardless of the directionality relation between ~Bmw and
the static field on account of quantum selection rules (see
[3, p. 49]).
With knowledge of the theory considered in sec-
tions II B and II C, the principles behind evaporative
cooling are in fact quite simple. In general, atoms trapped
in the |2, 2〉 state near the center of a magnetic quadrupole
trap have increasing energy with distance from the mag-
netic field’s minimum strength region. Microwaves de-
tuned a small frequency γ above the zero-field 6.834GHz
transition will interact only with atom’s whose Zeeman
splittings are large enough to reach the transition energy
corresponding to this raised frequency. This creates an
effective ‘edge’ to the trap, since atoms distanced farther
from the trap center are in higher magnetic fields, and
given enough kinetic energy may continue travelling until
their Zeeman splittings are resonant with hγ transitioning
them to an untrapped state. This is key: only the atoms
with enough kinetic energy, i.e. the hottest atoms, are
able to move into high enough fields to be freed from
the trap. It is easy to see, then, where this cooling
technique’s name comes from; perspiration, for example,
involves especially the hottest sweat evaporating into air,
carrying excess heat away and cooling the skin.
Keeping this in mind, the general procedure for evap-
orative cooling is as follows: Microwaves propagating
through the quadrupole trap are tuned to high enough
frequencies so as to only remove the highest temperature
trapped atoms. After rethermalizing, the remaining
sample’s temperature has been lowered. Repeating this
procedure at progressively lower frequencies can, in the-
ory, lower the temperature indefinitely [15].
In reality, this procedure is limited by multiple fac-
tors. Firstly, the collision rate between particles is
approximately nσvT where n is number density, σ is the
collision cross section, and vT =√
8kBT/(πM) is the
mean particle speed according to a Maxwell-Boltzmann
distribution. As this rate decreases proportionally to
T1/2 , the rethermalization rate becomes slower and slower
as the sample is cooled. This, combined with the limited
lifetime of a trapped sample, is a hindrance to reaching
quantum degeneracy temperatures. As a magnetic field of
the form in eq. (4) restricts colder atoms to move within
a smaller region of space near its minimum, it has the
counteracting effect of increasing density with decreasing
temperature. In a well designed cooling apparatus, the
latter effect may exceed the former such that rethermal-
ization rate increases with decreasing temperature, but
another effect sets a lower limit to evaporative cooling:
inelastic collisions. Internal energy may be transferred
between atoms in an inelastic collision, which in turn
can cause transitions from trapped to untrapped states
or simply an increase in an increase in one atom’s speed.
Overall, this gives evaporative cooling a lower bound on
the order of 1nK [15].
III. EXPERIMENTAL SETUP & PROCEDURE
Initially, a vapor of 87Rubidium atoms is doppler-
cooled by a 2-dimensional Magneto-Optical Trap (MOT)
produced by four lasers directed inward along two perpen-
dicular axes, in conjunction with four coils used as a 2D
magnetic quadrupole trap. As described in Section II A,
any transfer of momentum from the laser beams to the
vapor serves to cool the atoms. Another benefit of the
magnetic field gradient is therefore that, in addition to
trapping atoms, it serves to increase the likelihood of
momentum transfer from the laser beams. This is because
atoms moving outwards from the trap center may reach
the resonant Doppler shift for transitions more quickly as
a result of Zeeman splitting. As the trap potential and
laser beams are directed only along two dimensions, the
atoms retain their speed along the other perpendicular
axis and are therefore focused into a beam directed
towards the next stage of cooling.
A second MOT, established by three Helmholtz pairs
and lasers directed along all three perpendicular axes,
collects and confines the atoms in three dimensions.
This is where the majority of Doppler cooling occurs.
Afterwards, the magnetic trap is turned off, and optical
molasses cooling8 is performed for 18ms, then atoms are
optically pumped into the |2, 2〉 state over the course of
8This method uses detuned lasers and the Doppler effect to create
a frictional force approximately proportional to atomic velocity.
Though it does not provide an equilibrium restoring force, and
hence does not trap atoms, their ‘molasses’ like movement under
this friction force allows them to be held for the duration of this
process [? ].
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Christian Prosko (1353554) PHYS 499 - Final Report
FIG. 3. Evaporative cooling apparatus. The stub tuner allows for impedance matching, optimizing power transfer to waveguide.
Also, the circulator-isolator functions by permitting power transmission along one direction only, preventing interference of
reflected waves with other instruments.
1.2ms. The magnetic field is turned back on, and strength
is then ramped up for 1s, since the resulting higher
density increases rethermalization rates in preparation for
evaporative cooling.
Two primary experiments were attempted with our
microwave cooling apparatus, as illustrated in Figure 3.
In the former, the microwave generator was left on for
the entire course of the cooling cycle. The end of the
waveguide was positioned approximately 10cm from 87Rb
in the 3D quadrupole trap. Though it was also directed
towards the 2D MOT chamber, the microwaves would
have negligible interference with that stage as a result of
the rapid spatial decay of electromagnetic radiation. The
SG384 microwave generator was set to oscillate linearly
0.5MHz about a central frequency (at a rate of 5Hz) while
the Rubidium cloud is held in the trap for 5s, then the
trap was suddenly released, allowing the Rubidium cloud
to expand for 3ms before capturing an image. The entire
cycle was repeated 5 times for each central frequency,
after which the frequency was stepped down by 1MHz
and the procedure began again.
This allowed for observation of spin-flip transitions by
comparing the number of atoms in the cloud after 5
seconds with no microwaves, with that after 5 seconds
accompanied by a strong microwave signal. A high power
signal, however, incites more complete Rabi oscillations
over a wider range of frequencies (as shown in eq. (12)),
and therefore is expected to destroy the trapped vapor at
considerably higher frequencies than those for a low power
signal. To test this, a second round of data was collected
with a DC power source mixed into the microwave signal,
set to low enough power to attenuate the original signal
to about half its original amplitude.
To estimate the number of atoms in the cloud, ab-
sorption imaging is used. A laser tuned to a resonant
transition (780nm in this case [11], from 52S1/2 to 52P3/2)
passes through the trap when there are no atoms, and its
intensity I0 is measured by a camera. Another picture is
then taken after the cloud expansion, and the intensity of
images is compared according to the standard attenuation
formula:
I = I0e−σ
∫ndz (13)
where n is number density and σ in this case is the
scattering cross section for resonant light at the 780nm
transition. For distinguishable particles above ‘quantum’
temperatures, density is given by the Boltzmann distri-
bution:
n = n0e−H/(kBT ) (14)
for some n0. For a rough measure of particle number, we
approximate our Hamiltonian as being that of a simple
harmonic oscillator: H ≈ p2
2m+ 12mω
2r2 for some ω. Then
the number of atoms N in the cloud is about:
N =
∫ndxdydz = n0
(2πkBT
mω2
)3/2
(15)
Carrying out the integral in eq. (13) and substituting in
the result from eq. (15), we have:
− log
(I
I0
)= σN
(mω2
2πkBT
)e−(x2+y2)mω2/(2kBT ) (16)
≡ Ae−(x2+y2)/(2σxσy) (17)
Thus, to measure particle number, we numerically fit
the processed camera images to a Gaussian, and calculate
N from the resulting amplitude and standard deviations
according to:
N =2πσxσyA
σ(18)
For the optical transition in question, σ = 3λ2/(2π) ≈2.9× 10−13m−2.
The second experiment was an attempt in evaporative
cooling. After applying various frequency ramps, we
observed the atomic cloud’s size and compared it to
the cloud size with a static trapping field only. This
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Christian Prosko (1353554) PHYS 499 - Final Report
FIG. 4. Particles observed still inside trap after a time-of-
flight period of 3ms, as a function of trap edge (microwave
frequency). The uncertainty is approximately ±1 standard
deviation across values observed at the same frequency. Back-
ground denotes what the average absorption number would be
if eq. (15) was applied to background data (no atoms)
is because according to eq. (16), σx and σy are both
proportional to√T , so that a smaller cloud is signature of
lower temperature. Here, the microwave signal is mixed
with a radio-frequency signal triggered precisely after the
compression phase of the cycle, and turns off immediately
before the cloud is allowed to expand for imaging.
IV. RESULTS
Evidenced in Figure 4, the microwave apparatus pos-
sessed more than sufficient power to transition atoms out
of the trap. As expected from our discussion of Zeeman
splitting, this effect increased as the microwave frequency
approached resonance with the magnetically perturbed
hyperfine energy levels from above, and then decreased
just before the 6.834GHz limit was reached. The latter
effect is expected, since the magnetic field is non-zero
(almost) everywhere, so that essentially all atoms would
have energies slightly shifted from the unperturbed hy-
perfine transition. Also, our results were too inaccurate
to determine with certainty if a stronger field produced
a sharper ‘critical’ frequency after which atoms are more
rapidly removed. On the other hand, as the absorption
number for the high power setup remained within error
of 0 for 6.836 − 6.837GHz, the low power setup already
FIG. 5. Efficacy of evaporative cooling for two linear 5s ramps
at various microwave amplitude levels. x-axis is simply the
attenuation when compared to the unmodified RF signal.
was too far detuned from resonant energies to remove the
majority of atoms with these frequencies, in agreement
with the amplitude dependence of transition probabilities
(i.e. the squared coefficients from eqs. (11a) and (11b)).
Before applying our apparatus to evaporative cooling,
a suitable frequency ramp should be determined with
consideration of the microwaves’ intensity, range, and
the duration of the ramp. As such, linear ramps at-
tenuated by controlling the amplitude of the mixed RF
signal to various strengths were tried, and plotted in
Figure 5. Because an evaporative cooler seeks to lower
the temperature of a sample rather than eliminate its
presence completely, it is inferred from the data that
average powers greater than -10dBm would leave no
sample behind after cooling. Next, ramp duration was
varied for a single frequency range and power, with the
results displayed in Figure 6. Clearly, only ramps lasting
between 2 and 6 seconds were long enough to remove any
atoms, while short enough to be well within the trap’s
lifetime. It is worth noting that, with all experiments
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Christian Prosko (1353554) PHYS 499 - Final Report
FIG. 6. Efficacy of evaporative cooling for a single linear ramp,
spread over different ramp durations.
FIG. 7. The microwave frequency is lowered from frequencies
far detuned from 6.834GHz, displaying no loss of atoms.
Ramp Abs. Number N σxσy (m2)
None, 3.5s (6.9 ± 1.5) · 106 (3.5 ± 0.4) · 10−7
Linear,(3.6 ± 1.6) · 106 (5.9 ± 2.2) · 10−7
6.857-6.837GHz
Exponential,(6.1 ± 2.2) · 106 (3.8 ± 1.1) · 10−7
6.885-6.835GHz
TABLE I. Efficiency of cooling for linear ramps, and a ramp
with exponential decrease in frequency with respect to time.
Both above ramps lasted 3.5s. Data is averaged over 10 trials
for the first and last sweeps, and 5 trials for the second.
performed, uncertainty in the cloud size and absorption
number becomes very large for small values of these
quantities as a result of the breakdown of approximation
that the vapor’s density distribution is Gaussian.
Attempting evaporative cooling with different fre-
quency ranges and shapes of ramps suggested a critical
flaw in our experimental setup: the microwave ramps
consistently increased the cloud size, apparently heating
the atoms rather than cooling them. Some results are
presented in Table I, though many more ramps were
tested. Of course, it would be naive to assume that
the optimum microwave power and frequency change rate
should be the same at different frequencies, which is why
12 different ramps with step-wise exponentially decaying
frequency (at various powers) were tested in addition to
the 25 or so different linear ramps considered. The results
for the linear ramp in table I were typical of all linear
ramps attempted: All were able to effectively remove
atoms while leaving a small sample behind, though the
atomic vapor was heated in the process.
The exponentially decaying ramp is shown because it is
the only tested frequency sweep which at least maintained
a cloud size comparable to that of a pure magnetic trap
in the absence of microwaves, while still removing a
significant number of atoms. To be precise, this ramp
consisted of 500ms going from 6.885-6.855GHz, 500ms
from 6.855-6.845GHz, 500ms from 6.845-6.84GHz, and
finally 2s sweeping from 6.84-6.835GHz. The amplitude
of its RF-signal was set at each step to -10dBm, -15dBm,
-15dBm, and -20dBm, respectively. Its upper frequency
was determined by lowering the frequency from far above
ω0 until the absorption number slightly decreased (see
fig. 7).
8
Christian Prosko (1353554) PHYS 499 - Final Report
Unfortunately, none of these optimization efforts re-
vealed what was adding thermal energy to the system,
preventing evaporative cooling from being observed. It
is possible, however, that one source of heating is in-
terference of the unmixed microwave signal with the
trapped atoms. The microwave generator was set to
6.825GHz for the ramping experiments, a frequency that
may not have been detuned enough from 6.834GHz to
prevent interactions with the trap. Though the mixer
used weakens the unmixed signal, it still remained at
more than 50% percent of the mixed signal’s amplitude,
as measured by a frequency analyzer tapped into the
circuit through a directional coupler. On the other hand,
absorption numbers observed with the frequency ramp
turned off are in the same range as those with the ramp
turned on, but with a high cutoff frequency. If any atoms
were leaking through this possible ‘hole’ in the trap,
there were unmeasurably few. Nonetheless, colder atoms
are more likely to be located near the trap minimum,
where the 6.825GHz signal is least detuned from Zeeman
split energies. Thus, it is possible that a sort of reverse
evaporative cooling was occurring, with primarily the
coldest atoms escaping the trap through this interaction.
V. CONCLUSION
The intrinsically quantum mechanical effects of os-
cillating magnetic fields on atomic angular momentum
were demonstrated by utilizing microwaves to ‘flip’ 87Rb
atoms from magnetically trapped total angular momen-
tum states to untrapped states. These observations
are explained by weak-coupling perturbation theory of a
magnetic field with magnetic sublevels of the atoms. Un-
fortunately, an unforeseen heating mechanism prevented
the observation of evaporative cooling, although lowering
the frequency of the unmixed microwave signal could
remedy this.
Appendix A: Derivation of the Breit-Rabi Formula
If an alkali atom is in its fine structure ground state
such that L = 0, J = 1/2, and I = 3/2, the most complete
hydrogenic basis states are |mI ,mJ〉. Under these con-
ditions, the unperturbed Hamiltonian is effectively the
hyperfine coupling between electron and nuclear angular
momentum, for which |F,mF 〉 are eigenstates:
H = AI · J + (µI + µJ) · ~B (A1)
where A is an energy constant for a given isotope and L
level. Using the standard ladder operators Q± ≡ Qx ±iQy, we may rewrite this as:
H = A(Ix + Iy + Iz) · (Jx + Jy + Jz) + µBB(gI Iz + gJ Jz)
= A
[IzJz +
(I+ + I−
2
)(J+ + J−
2
)
+i
2(I− − I+)
i
2(J− − J+)
]+ µBB(gI Iz + gJ Jz)
= A
[IzJz +
1
2(I+J− + I−J+)
]+ µBB(gI Iz + gJ Jz)
(A2)
In this system, mF is conserved such that mF = mI+mJ .
Any suspicion towards the seemingly classical treatment
of the dipole moment dot product with ~B is well founded,
as I and J dipole moments are in general precessing about
the quantization axis in possibly different directions.
This is called ‘Larmor Precession,’ and its behaviour is
accounted for by the g-factors. Thus, we approximate the
perturbed energies by diagonalizing the Hamiltonian’s
matrix form in the |mJ = ±1/2,mI = mF ∓ 1/2〉 ≡ |±〉basis. Its elements are calculated below for given mF :
〈±|H |±〉 = A 〈±| IzJz |±〉
+A
2
[〈±| I+J− |±〉+ 〈±| I−J+ |±〉
]+ µBB
[gJ 〈±| Jz |±〉+ gI 〈±| Iz |±〉
]= AmJmI +
A
2[0 + 0] + µBB [gJmJ + gImI ]
= A(±1/2)(mF ∓ 1/2) + µBB
[±1
2gJ + (mF ∓
1
2)gI
]= −A
4± AmF
2+ µBBgImF ±
µBB
2(gJ − gI) (A3)
We have used the the property that Q± |Q,mQ〉 =√Q(Q+ 1)−mQ(mQ ± 1) |Q,mQ ± 1〉 returns 0 when
|mQ| would exceed Q as a result. No h appears upon
taking inner products simply because it is contained in
the definitions of A and µB . Similarly, for 〈±| H |∓〉:
〈+| H |−〉 =A
2
[〈+| I+J− |−〉+ 〈+| I−J+ |−〉
]=A
2
[0 +
√(I +mF + 1/2)(I −mF + 1− 1/2)
×√
(1/2 + 1/2)(1/2 − 1/2 + 1)]
=A
2
√(I −mF + 1/2)(I +mF + 1/2)
=A
2
√(I + 1/2)2 −m2
F (A4)
9
Christian Prosko (1353554) PHYS 499 - Final Report
which is also the expression for 〈−| H |+〉 since H is
Hermitian. Thus, our Hamiltonian matrix has the form:
H =
(〈+| H |+〉 〈+| H |−〉〈−| H |+〉 〈−| H |−〉
)=
(a+ b c
c a− b
)(A5)
Setting det(H − E) = 0 we find the eigenvalue equation
E2 − 2aE + a2 − b2 − c2 = 0, which may be solved using
the quadratic formula. This yields E = a±√b2 + c2 from
which the Breit-Rabi formula follows after some trivial
cancellations.
Appendix B: Rabi Flopping Matrix Elements
In this section, we calculate Ω for the |2, 2〉 → |1, 1〉transition as an illustration of how similar matrix ele-
ments may be found. We may expand |F,mF 〉 states in
terms of |mI ,mS〉 states in the usual fashion, that is, by
introducing an identity operator∑|mI ,mS〉 〈mI ,mS | to
the right hand side:
|2, 2〉F = |3/2,1/2〉 〈3/2,
1/2|2, 2〉F= |3/2,
1/2〉 (B1)
|1, 1〉F = |3/2,−1/2〉 〈3/2,−1/2|1, 1〉F+ |1/2,
1/2〉 〈1/2,1/2|1, 1〉F
=
√3
4|3/2,−1/2〉 −
1
2|1/2,
1/2〉 (B2)
where an F subscript denotes the |F,mF 〉 basis, and
|mI ,mS〉 is implied otherwise. Inner products in the
above equations are Clebsch-Gordan coefficients [5, p.
188], which are zero when mF 6= mI + mS , hence there
only being one or two terms in each expansion. Next, we
note that to calculate Ix |mI〉 or Sx |mS〉, we must use the
property that Qx = 12 (Q+ + Q−), recalling the properties
of Q± as mentioned in appendix A:
Ω =µBB
mw
2h〈3/2,
1/2|[gI(I+ + I−) + gS(S+ + S−)
]×
[√3
4|3/2,−1/2〉 −
1
2|1/2,
1/2〉
]
=µBB
mw
2h
√3
2(gS − gI) (B3)
≈ (7.62853 · 1010Hz/T)µBBmw (B4)
We have used the values gS ≈ 2.00319, gI ≈ −0.000995.
Most of the terms in the above calculation cancel
by the orthonormality relation 〈mI ,mS |m′I ,m′S〉 =
δmI ,m′IδmS ,m′
S.
10
Christian Prosko (1353554) PHYS 499 - Final Report
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